Giovanni P. Crespi, Ivan Ginchev, Matteo Rocca. Variational inequalities in vector optimization 2004/31

Transcription

1 Giovanni P. Crespi, Ivan Ginchev, Matteo Rocca Variational inequalities in vector optimization 2004/31 UNIVERSITÀ DELL'INSUBRIA FACOLTÀ DI ECONOMIA

2 In questi quaderni vengono pubblicati i lavori dei docenti della Facoltà di Economia dell Università dell Insubria. La pubblicazione di contributi di altri studiosi, che abbiano un rapporto didattico o scientifico stabile con la Facoltà, può essere proposta da un professore della Facoltà, dopo che il contributo sia stato discusso pubblicamente. Il nome del proponente è riportato in nota all'articolo. I punti di vista espressi nei quaderni della Facoltà di Economia riflettono unicamente le opinioni degli autori, e non rispecchiano necessariamente quelli della Facoltà di Economia dell'università dell'insubria. These Working papers collect the work of the Faculty of Economics of the University of Insubria. The publication of work by other Authors can be proposed by a member of the Faculty, provided that the paper has been presented in public. The name of the proposer is reported in a footnote. The views expressed in the Working papers reflect the opinions of the Authors only, and not necessarily the ones of the Economics Faculty of the University of Insubria. Copyright Ivan Ginchev, Giovanni P. Crespi, Matteo Rocca Printed in Italy in October 2004 Università degli Studi dell'insubria Via Ravasi 2, Varese, Italy All rights reserved. No part of this paper may be reproduced in any form without permission of the Author.

3 Variational inequalities in vector optimization Giovanni P. Crespi Ivan Ginchev Matteo Rocca Abstract In this paper we investigate the links among generalized scalar variational inequalities of differential type, vector variational inequalities and vector optimization problems. The considered scalar variational inequalities are obtained through a nonlinear scalarization by means of the so called oriented distance function [14, 15]. In the case of Stampacchia-type variational inequalities, the solutions of the proposed ones coincide with the solutions of the vector variational inequalities introduced by Giannessi [8]. For Minty-type variational inequalities, analogous coincidence happens under convexity hypotheses. Furthermore, the considered variational inequalities reveal useful in filling a gap between scalar and vector variational inequalities. Namely, in the scalar case Minty variational inequalities of differential type represent a sufficient optimality condition without additional assumptions, while in the vector case the convexity hypothesis was needed. Moreover it is shown that vector functions admitting a solution of the proposed Minty variational inequality enjoy some well-posedness properties, analogously to the scalar case [4]. 1 Introduction Given a map F from R n to R n and a nonempty set K R n, we say that a point x K is a solution of a Stampacchia variational inequality when [13]: VI(F, K) F (x ),y x 0, y K. Analogously we say that x K is a solution of a Minty variational inequality when [17]: MVI(F, K) F (y),x y 0, y K. Université delavallée d Aoste, Facoltà di Economia, Strada dei Cappuccini 2A, Aosta, Italia. e mail: g.crespi@univda.it Technical University of Varna, Department of Mathematics, Studentska str., 9010, Varna, Bulgaria. e mail: ginchev@ms3.tu-varna.acad.bg Università dell Insubria, Dipartimento di Economia, via Ravasi 2, Varese, Italia. e mail: mrocca@eco.uninsubria.it 1

4 In particular, when the variational inequality admits a primitive minimization problem (that is the function f to minimize is such that F = f )andk is a convex set, VI(f,K)andMVI(f,K) have strong links with this problem. Roughly speaking, VI(f,K) is a necessary condition for the minimization of the function f over the set K, which becomes also sufficient when f is convex. On the contrary, MVI(f,K) is a sufficient condition for the minimization of f over the set K, which becomes necessary if f is convex. Recently it has been observed also [4] that the existence of a solution of MVI(f,K) has some implications on the well-posedness of the related optimization problem. Variational inequalities in the sense of Minty and Stampacchia have been extended to the case where F is a point to set map from R n 2 Rn (see for instance [10]). In this case a point x K is a solution of a Stampacchia variational inequality, when there exists ξ F (x ), such that ξ,y x 0, y K. Analogously x K is said a solution of a Minty variational inequality when it holds v, x y 0, y K and v F (y). Furthermore a vector extension of Minty and Stampacchia variational inequalities has been introduced by F. Giannessi [8, 9], who has also given some links between the solutions of vector variational inequalities and the solutions of a vector optimization problem. Roughly speaking, also for the vector case it has been proved that Stampacchia vector variational inequalities represent a necessary condition for optimality (that becomes sufficient under convexity assumptions). Analogously to the scalar case it is proved that Minty vector variational inequality is a necessary and sufficient optimality condition under convexity assumptions. But a gap with the scalar case arises, namely that convexity is needed also to prove that Minty vector variational inequality is a sufficient optimality condition. In this paper we introduce a generalization of scalar variational inequalities (of differential type) and we investigate their links with vector variational inequalities and vector optimization problems. The considered variational inequalities are obtained through a nonlinear scalarization, which makes use of the so called oriented distance function [14, 15]. We show that the solutions of the proposed variational inequalities coincide with the solutions of some variational inequalities for point to set maps. In the case of Stampacchia-type variational inequalities the links with vector optimization coincide with those holding for vector valued ones. For Minty-type variational inequalities analogous coincidence holds under convexity assumptions. We show that if the convexity hypothesis is dropped, the proposed Minty variational inequalities provide a stronger solution concept with respect to Minty vector variational inequalities and are useful in filling the previously mentioned gap. Moreover it is shown that vector functions admitting a solution of the proposed Minty variational inequality enjoy well-posedness properties analogously to the scalar case [4]. The paper is structured as follows. In section 2 we recall some known results about Minty variational inequalities and scalar optimization. Section 3 presents the concept of oriented distance function and its application in the scalarization of vector optimality concepts. Section 4 deals with variational inequalities and vector opti- 2

5 mization. 2 Scalar variational inequalities We are concerned with the following optimization problem: P (φ, K) min φ(x), x K x K Rn, where φ : R n R. A point x K is a solution of P (φ, K) whenφ(x) φ(x ) 0, x K. The solution is strong when φ(x) φ(x ) > 0, x K\{x }. In this section we assume that φ is a fuction defined and directionally differentiable on an open set containing K. We recall that the directional derivative of φ at apointx in the direction d R n is defined as: φ φ(x + td) φ(x) (x; d) = lim, t 0 + t when this limit exists and is finite. We deal with the following variational problems: VI(φ,K) Find a point x K such that φ (x ; y x ) 0, y K. MVI(φ,K) Find a point x K such that φ (y; x y) 0, y K. Observe that the previous problems reduce to the classical Stampacchia and Minty variational inequalities when φ is differentiable, which induces us to use the classical abbreviations VI and MVI. Definition 1. i) Let K be a nonempty subset of R n. The set ker K consisting of all x K such that (y K, t [0, 1]) = x + t(y x) K is called the kernel of K. ii) A nonempty set K is star-shaped if ker K. In the following we will use the abbreviation st-sh for the word star-shaped. It is known (see e.g. [18]) that the set ker K is convex for an arbitrary st-sh set K. We will assume, by definition, that the empty set is st-sh. Definition 2. Afunctionφ defined on R n is called increasing along rays at a point x (for short, f IAR(x )) if the restriction of this function on the ray R x,x = {x + αx α 0} is increasing for each x R n.(afunctiong of one real variable is called increasing if t 2 t 1 implies g(t 2 ) g(t 1 ).) Definition 3. Let K R n be a st-sh set and x ker K. Afunctionφ defined on K is called increasing along rays at x (for short, φ IAR(K, x )), if the restriction of this function on the intersection R x,x K is increasing, for each x K. 3

6 Proposition 1. [4] i) If φ IAR(K, x ),thenx is a solution of P (φ, K). ii) φ IAR(K, x ) if and only if x lev c φ := {x K φ(x) c}). ker lev c φ for every c φ(x ) (here The following result can be deduced from Theorem 2 in [4] Proposition 2. i) Let x be a solution of MVI(φ,K) and x ker K. Then φ IAR(K, x ). ii) Let φ IAR(K, x ).Thenx is a solution of MVI(φ,K). Remark 1. If x isastrongsolutionofmvi(φ,k) (i.e. φ (y; x y) < 0, y K\{x }), in the previous Proposition we can easily conclude with the same proof that φ is strictly increasing along rays starting at x. The following result has an immediate proof and we omit it. Proposition 3. i) Let x ker K. If x K is a solution of P (φ, K), thenx solves VI(φ,K). ii) Let K be a convex set. If φ is convex and x K solves VI(φ,K) then x is asolutionofp (φ, K). Proposition 4. i) Let x ker K. Ifx K is a (strong) solution of MVI(φ,K) then x is a (strong) solution of P (φ, K). ii) Let K be a convex set. If x K solves P (φ, K) and φ is convex, then x solves MVI(φ, K). Proof: i) Since x is a solution of MVI(φ,K), then φ IAR(K, x ) and hence x solves P (φ, K). Analogously when x isastrongsolutionofmvi(φ,k). ii) If φ is convex and x K solves P (φ, K), then φ IAR(K, x )andsox solves MVI(φ,K). Problems VI(φ,K)andMVI(φ,K) can be linked by the following result analogous to the classical Minty s Lemma. Proposition 5. i) Let x ker K. Ifx K solves MVI(φ,K) and φ ( ; d) is upper semicontinuous (u.s.c.) along rays starting at x for every d R n,then x is a solution of VI(φ,K). 4

7 ii) Let K be a convex set. If x K solves VI(φ,K) and φ is convex, then x solves MVI(φ,K). Proof: i) We begin proving that under the assumptions, if x K solves MVI(φ,K), then x is such that φ (y; y x ) 0, y K. Sincex solves MVI(φ,K), we know that φ IAR(K, x ) and since x ker K, theset{r x,y K} is convex and hence has a nonempty relative interior ri {R x,y K}. Ify ri {R x,y K}, for t>0 small enough, we have y+t(y x )=x +(1+t)(y x ) R x,y K and hence φ(y+t(y x )) φ(y), from which it follows easily φ (y; y x ) 0. Let now y {R x,y K}\ri {R x,y K}. Hence we have y = lim y k,forsome sequence y k ri {R x,y K}, thatisy k = x + t k (y x ). It holds: and hence: 0 φ (y k ; y k x )=φ (x + t k (y x ); t k (y x )) 0 lim sup φ (x + t k (y x ); y x ) φ (y; y x ), k + where the last inequality follows since φ (,d) is u.s.c. along rays starting at x. Let now z K and consider the point z(t) :=x +t(z x ), t (0, 1]. We have 0 φ (z(t); z(t) x )=φ (z(t); t(z x )) and hence φ (z(t); (z x )) 0. Passing to the limit as t 0 + and taking into account the fact that φ ( ; y x ) is u.s.c. along rays starting at x we get φ (x ; y x ) 0. ii) If x K solves VI(φ,K), then x solves P (φ, K) andφ IAR(K, x ). Hence x solves MVI(φ,K). Now we recall the notion of Tykhonov well-posedness for problem P (φ, K). Definition 4. Asequencex k K is a minimizing sequence for P (φ, K), when φ(x ) inf K φ(x). Definition 5. Problem P (φ, K) is Tykhonov well-posed when it admits a unique solution x and every minimizing sequence for P (φ, K) converges to x. For ε>0weset: L φ (ε) ={x K : φ(x) inf φ + ε} K and we recall the following result (see e.g. [7]). Theorem 1. i) If P (φ, K) is Tykhonov well-posed, then diam L φ (ε) 0 as ε 0 +, or equivalently inf ε>0 diam L φ (ε) =0(here diam A denotes the diameter of the set A). 5

8 ii) Let φ be lower semicontinuous and bounded from below. If inf ε>0 diam L φ (ε) = 0, then P (f,k) is Tykhonov well-posed. Theorem 2. [4] Let K beaclosedsubsetofr n, x ker K and f IAR(K, x ). If P (φ, K) admits a unique solution, then it is Tykhonov well-posed. 3 Scalar characterizations of vector optimality concepts Let C be a closed, convex, pointed cone with nonempty interior. Let M be any of the cones C c, C\ { 0 }, C and int C. The vector optimization problem (see e.g. [19]) corresponding to M, wheref : R n R l, is written as: VP(f,K) v min M f(x), x K. This amounts to find a point x K (called the optimal solution), such that there is no y K\{x } with f(y) f(x ) M. The optimal solutions of the vector problem corresponding to C c (respectively, C\{0}, C and int C) are called ideal solutions (respectively, efficient solutions, strongly efficient solutions and weakly efficient solutions). We will denote the efficient solutions as e-solutions and the weakly efficient solutions as w-solutions. Let us now recall the notion of oriented distance function, introduced by Hiriart-Hurruty [14, 15]. Definition 6. For a set A R l let A : R l R {± } be defined as: A (y) =d A (y) d R l \A(y), where d A (y) :=inf a A y a is the distance from the point y to the set A. Wewill call function A (y), the oriented distance from the point y to the set A. Function A has been recently used in [20] to characterize several notions of efficient point of a given set D R l. In [12] it has been proved that when A is a closed, convex, pointed cone, then we have: A (y) = max ξ A S ξ,y, where A := {x R l x, a 0, a A} is the positive polar of the set A and S the unit sphere in R l. In this section we use function A in order to give scalar characterizations of some notions of efficiency for problem VP(f,K). Furthermore, some results characterize pointwise well-posedness of problem VP(f,K) [6] through function A. Given a point ˆx K, consider the function: φˆx (x) = max ξ,f(x) f(ˆx), ξ C S 6

9 where C denotes the positive polar of C and S the unit sphere in R l. φˆx (x) = C (f(x) f(ˆx)). We consider the problem: Clearly P (φˆx,k) minφˆx (x), x K. The following Theorem can be found in [11]. Theorem 3. i) The point x K is a strong e-solution of VP(f,K) if and only if x is a strong solution of P (φ x,k). ii) The point x K is a w-solution of VP(f,K) if and only if x is a solution of P (φ x,k). The next result slightly extends Theorem 3. Theorem 4. i) The point x K is a strong e-solution of VP(f,K) if and only if there exists a point ˆx K, such that x is a strong solution of P (φˆx,k). ii) The point x K is a w-solution of VP(f,K) if and only if there exists a point ˆx K, such that x is a solution of P (φˆx,k). Proof. We prove only i), since the proof of ii) is analogous. Let x be a strong e-solution of VP(f,K). Then from Theorem 3 we know that x is a strong solution of P (φ x,k) and necessity is proved. Now, assume that for some ˆx K, x is a strong solution of P (φˆx,k), i.e. φˆx (x ) <φˆx (x), x K\{x }, or equivalently: max ξ C S ξ,f(x ) f(ˆx) < max ξ,f(x) f(ˆx) = ξ C S max ξ,f(x) ξ C S f(x )+f(x ) f(ˆx) max ξ,f(x) ξ C S f(x ) + max ξ C S ξ,f(x ) f(ˆx), x K\{x }. Hence max ξ C S ξ,f(x) f(x ) > 0, x K\{x }, i.e. x isastrongsolutionof P (φ x (x),k). From the previous Theorem we obtain that x is a strong e-solution of VP(f,K). Now we recall the notion of pointwise well-posedness for problem VP(f,K) [6]. Let k C, α>0, v K and set: L(v, k, α) ={x K f(x) f(v)+αk C}. Definition 7. Problem VP(f,K) is said to be pointwise well-posed at the e-solution x when: inf α>0 diam L(x,k,α)=0, for each k C. 7

10 Theorem 5. Let f be a continuous function and let x K be an e-solution of VP(f,K). Problem VP(f,K) is pointwise well-posed at x if and only if problem P (φ x,k) is Tykhonov well-posed. Proof. Since x is an e-solution of VP(f,K), then x is also a w-solution of VP(f,K) and hence (Theorem 3) a solution of P (φ x,k), with φ x (x )=0. LetP (φ x,k) be Tykhonov well-posed. If for some k C and α>0, x L(x,k,α), then, for some c C, wehavef(x) f(x )= c + αk and so: φ x (x) = max ξ,f(x) ξ C S f(x ) = max ξ, c + αk ξ C S max ξ, c + α max ξ,k α max ξ,k. ξ C S ξ C S ξ C S (the last inequality follows since for every ξ C S, wehave ξ, c 0). Hence we have x L φ x (α max ξ C S ξ,k ). It follows that α >0and k C, wehave: and so, k C: L(x,k,α) L φ x (α max ξ C S ξ,k ) inf α>0 diaml(x,k,α) inf diam α>0 Lφ x (α max ξ,k ). ξ C S Since P (φ x,k) is Tykhonov well-posed, we have: inf diam α>0 Lφ x (α max ξ,k ) =0 ξ C S and hence inf α>0 diam L(x,k,α)) = 0, that is VP(f,K) is pointwise well-posed at x. Assume now that VP(f,K) is pointwise well-posed at x. We prove that there exists apoint k int C such that for every α>0itholds: L φ x (α) L(x, k, α). For every k int C and ξ C S we have ξ,k > 0. Choose a vector k int C with min ξ C S ξ, k > 1. If, ab absurdo, for some α > 0thereexistsapoint x L φ x (α)\l(x, k,α), then we have f(x) f(x ) C + α k. It follows the existence of a point ξ C S such that ξ,f(x) f(x ) α k > 0andso: from which: ξ,f(x) f(x ) >α ξ, k, max ξ,f(x) ξ C S f(x ) >α ξ, k α min ξ, k >α, ξ C S that is φ x (x) φ x (x ) >αand hence the absurdo x L φ x (α). So we have: L φ x (α) L(x, k,α), α >0. 8

11 Since VP(f,K) is pointwise well-posed at x,wehaveinf α>0 diam L(x, k,α) =0 and so also inf α>0 diam L φ x (α) = 0. Since f is continuous then also φˆx is continuous (see e.g. [18]) and so the proof is complete recalling ii) of Theorem 1. In the scalar case it is known that if φ is a convex function with a unique (strong) minimizer over K, then problem P (φ, K) is Tykhonov well-posed. Now we extend this property to the vector case. Definition 8. The function f : K R n R l is said to be C convex when: f (λx +(1 λ)y) [ λf (x)+(1 λ) f (y) ] C x, y K, λ [0, 1]. The following result has an almost immediate proof and we omit it. Proposition 6. Let f : R n R l be a C-convex function. function φˆx (x) is convex. Then ˆx K, the Theorem 6. If f : R n R l is C-convex, then f is pointwise well-posed at any strong e-solution of VP(f,k). Proof: Assume that f is C-convex and let x be a strong e-solution of VP(f,K). Then, from Theorem 3 x is the unique minimizer of the convex function φ x (x) over K and a classical result (see [7]) states that problem P (φ x,k) is Tykhonov well-posed. The thesis then follows from Theorem 5 Remark 2. If we consider C = R l + and define φˆx (x) =max{f i (x) f i (ˆx),i = 1,...,l}, then it can be proved [4] that φˆx (x) = φˆx (x). 4 Variational inequalities and vector optimization Vector variational inequalities (of Stampacchia type) have been first introduced in [8]. Later a vector formualtion of Minty variational inequality has been proposed as well (see e.g. [9]). Both the inequalities involve a matrix valued function F : R n R l n and a feasible region K R n. We consider the following sets: { } Ω(x) := u R l u = F (x)(y x), y K, Θ(x) := { } w R l w = F (y)(y x), y K. Definition 9. i) A vector x K is a solution of a strong vector variational inequality of Stampacchia type when: VVI s (F, K) Ω(x ) ( C ) = {0}. 9

12 ii) A vector x K is a solution of a weak vector variational inequality of Stampacchia type when: VVI(F, K) Ω(x ) ( int C ) =. Definition 10. i) A vector x K is a solution of a strong vector variational inequality of Minty type when: MVVI s (F, K) Θ(x ) ( C ) = {0}. ii) A vector x K is a solution of a weak vector variational inequality of Minty type when: MVVI(F, K) Θ(x ) ( int C ) =. In the sequel we will deal mainly with weak vector variational inequalities of Stampacchia and Minty type (for short VVI and MVVI, respectively). The following result (see [9]) extends the classical Minty s Lemma to the vector case. Lemma 1. Let K be a convex set and let F be hemicontinuous and C-monotone. Then x is a solution of MVVI(F, K) if and only if it solves VVI(F, K). Similarly to the scalar case, we consider a function f : R n R l, that we assume to be differentiable on an open set containing K, such that f = F, for all x K (here f denotes the Jacobian of f). The following results (see [8, 9]) link VVI(f,K)andMVVI(f,K) to vector optimization. Proposition 7. Let f : R n R l be differentiable on an open set containing K. i) If x ker K is a a w-solution of VP(f,K), then it solves also VVI(f,K). ii) If K is a convex set, f is C convex and x is a solution of VVI(f,K), then it is a w-solution of VP(f,K). Some refinements of the relations between VVI and efficiency have been given in [2]. Proposition 8. Let C = R l + and K be a convex set. If f is C convex and differentiable on an open set containing K, thenx K is a w-solution of VP(f,K) if and only if it is a solution of MVVI(f,K). Remark 3. The previous result has been extended to an arbitrary ordering cone C (closed, convex, pointed and with nonempty interior) in [3], under the hypothesis that f is hemicontinuous at x, i.e. that the restriction of f on any ray starting at x is continuous. This assumption in not really additional with respect to Proposition 8, since the Jacobian of every R l + -convex and differentiable function is hemicontinuous. 10

13 In particular, Proposition 8 gives an extension to the vector case of Proposition 4 (for differentiable functions). Anyway, in Proposition 8, convexity is needed also for proving that MVVI(f,K) is a sufficient condition for optimality, while in the scalar case, convexity is needed only in the proof of the necessary part. The next example shows that the convexity assumption in Proposition 8 cannot be dropped. Example 1. Let C = R 2 +, K = [ 2 π, 0] and consider a function f : R R 2, [ ] f1 (x) f(x) =, defined as follows. We set: f 2 (x) { x f 1 (x) = 2 sin 1 x x2, x 0 0, x =0 and observe that 2x 2 f 1 (x) 0, x K and f 1 is differentiable on K. Function f 1 has a countable number of local minimizers and of local maximizers over K. The local maximizers of f 1 are the points y k = 1 π +2kπ, k =0, 1,... and f 1(y k )=0.Ifwe 2 denote by x k, k =0, 1,... the local minimizers of f over K, we have y k <x k <y k+1, k =0, 1,.... Function f 2 is defined on K as: f 2 (x) = f 1(x k ) 2 f 1(x k+1 ) 2 [ cos ( πx x k y k + π(x k 2y k ) x k y k ) 1 [ cos ( πx y k+1 x k + π(2y k+1 3x k ) ], x [y k,x k ) ) ] y k+1 x k 1, x [x k,y k+1 ) 0, x =0 for k =0, 1,.... It is easily seen that also f 2 is differentiable on K. The graphs of f 1 and f 2 are plotted in figure 1. The points x [ 2 π,x 0] are w-solutions, while the other points in K are not w- solutions. In particular, x =0is an ideal maximal point (i.e. f(x) f(x ) R 2, x K). Anyway, it is easy to see that any point of K is a solution of MVVI(f,K). In order to fill the gap between Proposition 8 and the analogous scalar result, we consider function φˆx introduced in the previous section. From now on we will assume that f is a function of class C 1 on an open set containing K (this assumption can be weakened with differentiability when C = R l +). The following Theorem resumes some classical properties of function φˆx. Theorem 7. [5] i) φˆx is directionally differentiable and φ (x; d) = max ξ Rˆx (x) ξ f (x)d, where Rˆx (x) ={ξ C S : φˆx (x) = ξ,f(x) f(ˆx) }. 11

14 Figure 1: f 1 (x) andf 2 (x). ii) φ (x; ) is sublinear and can be expressed as: φ (x; d) = max v, d, v φˆx (x) where φˆx (x) =conv{ξ f (x), ξ Rˆx (x) } (here conv A denotes the convex hull of the set A). Now we consider the following problems: VI(φ,K) For 0, y K. a given ˆx K, find a point x K such that φ (x ; y x ) MVI(φ,K) For 0, y K. a given ˆx K, find a point x K such that φ (y; x y) Remark 4. Clearly, Proposition 5 prvides some links between these two problems. Since, under the made assumptions, φ (x; ) is u.s.c., then any solution of Problem MVI(φ,K) is a solution of VI(φ,K). Conversely, if f is C-convex, then φˆx is convex (see Proposition 6) and hence Proposition 5 states that every solution of Problem VI(φ,K) is also a solution of MVI(φ,K). The next results state the equivalence betweeen the previous problems and generalized variational inequalities for point to set maps [10]. Proposition 9. Let K be a convex set. Problem VI(φ,K) is equivalent to the following generalized variational inequality of Stampacchia type : 12

15 VI( φˆx,k) For some given ˆx K, find a point x K, such that v φˆx (x ) for which v, x y 0. Proof: VI( φˆx,k) = VI(φ,K) is obvious. Instead, assume that x solves VI(φ,K), i.e. φ (x ; y x ) 0, y K. This means: max v, y v φˆx (x ) x 0, y K and the result follows from Lemma 1 in [1]. Similarly we get the following result which we state without the obvious proof. Proposition 10. Let K be a convex set. Problem MVI(φ,K) is equivalent to the following generalized variational inequality of Minty type: MVI( φˆx,k) For a given ˆx K, find a point x K such that v, x y 0 for every v φˆx (y) and for every y K. Now we prove that the solutions of problem VI(φ,K) coincide with the solutions of VVI(f,K). Proposition 11. Let K be a convex set. If x K solves problem VI(φ,K) for some ˆx K, thenx is a solution of VVI(f,K). Conversely, if x K solves VVI(f,K), thenx solves problem VI(φ x,k). Proof: Assume first that x solves problem VI(φ,K)forsomeˆx K. Then from Proposition 9 we know that x solves VI( φˆx,k), i.e. there exists v φˆx (x ), such that v,y x 0, y K. By Caratheodory Theorem v = r i=1 λ iξi f (x ), with 0 <r n +1, λ i 0, r i=1 λ i =1,ξ i Rˆx (x ). This means r i=1 λ iξi f (x )(y x ) 0, y K. Ab absurdo assume that for some y K it holds f (x )(y x ) int C. Hence, for every ξ C S, wemusthave ξ f (x )(y x ) < 0 and this conradicts the previous inequality. Assume now that x K solves VVI(f,K) and observe that since K is convex, also Ω(x ) is a convex set. Since Ω(x ) int C =, then from the well known Separation Theorem, we have the existence of a vector ξ C S such that ξ f (x )(y x ) 0. Now,observethatwehaveφ x (x ; y x )=max ξ Rx (x ) ξ f (x )(y x )and R x (x )=C S. So the previous inequality implies φ x (x ; y x ) 0. Remark 5. In [16] it has been proved that, under the hypotheses of the previous result, the set of the solutions of VVI(f,K) coincide also with the set of the solutions of the scalar variational inequalities VI(ξ f,k), ξ C. Now we turn our attention to problem MVI(φ,K). Theorem 8. Let x K solve MVI(φ,K). Thenx solves MVVI(f,K). Proof: Let x solve MVI(φ,K) and ab absurdo assume that x does not solve MVVI(f,K). Hence, for some ȳ K we have f (ȳ)(ȳ x ) int C and so ξ f (ȳ)(ȳ x ) < 0, ξ C S. This contradicts the fact that x solves MVI(φ,K), i.e. that max ξ Rˆx (y) f (y)(y x ) 0, y K. 13

16 The converse of the previous result holds under convexity assumptions. Theorem 9. Let K be a convex set and f be a C-convex function. If x K solves MVVI(f,K), thenx solves problem MVI(φ x,k). Proof: We know that, if f is C-convex and x solves MVVI(f,K), then x is a w-solution of VP(f,K) (Proposition 8 and Remark 3) and hence x is a solution of P (φ x,k) (Theorem 3). Since f is C-convex, from Proposition 6 we know that φ x (x) isconvexandthenφ x IAR(K, x ). It follows that φ x (y; x y) 0 (recall Proposition 2) and the proof is complete. The convexity assumption in the previous result cannot be dropped as the following example shows. Hence, when convexity assumptions do not hold MVI(φˆx,K) defines a stronger solution concept then MVVI(f,K). Example 2. Consider the function of Example 1 that clearly is not R 2 + convex. The point x =0is a solution of MVI(f,K), butthereisnoˆx [ 2 π, 0] such that x solves MVI(φ,K), withφˆx(x) =max{f 1 (x),f 2 (x)} (recall Remark 2). Theorem 10. Let x ker K be a solution of MVI(φ,K) for some ˆx K. Then x is a w-solution of VP(f,K). Proof: Since x solves MVI(φ,K), then x is a solution of P (φˆx,k) and hence a w-solution of VP(f,K) (recall Proposition 4 and Theorem 4). Theorem 11. Let x ker K. If x is a strong solution of MVI(φ,K) for some ˆx K, thenx is a strong e-solution of VP(f,K). Furthermore, if x is a strong solution of MVI(φ x,k), thenvp(f,k) is pointwise well-posed at x. Proof: If x is a strong solution of problem MVI(φ,K)forsomeˆx, thenx is a strong e-solution of VP(f,K) (apply Propositions 4 and Theorem 4). Assume in particular that x is a strong solution of problem MVI(φ x,k), i.e.: φ x (y; x y) < 0, y K\{x }. Then combining Propositions 2 and 4 and Theorems 2 and 5, the proof is complete. Example 3. Consider the function f : R R 2 defined as f(x) =(x, log x 1 ) = (f 1 (x),f 2 (x)), letc = R 2 + and K =[ 1/2, 1/2]. It is easy to check that x =0solves MVVI(f,K) and x is a e-solution of VP(f,K) (and hence also a w-solution). Anyway, Proposition 8 would not have allowed such a conclusion, since f is not C- convex. Instead, considering function φ x (x) =max{f 1 (x),f 2 (x)} one gets that x is a strong solution of MVI(φ x,k) and hence x is a strong e-solution of VP(f,K). Furthermore VP(f,K) is pointwise well-posed at x. 14

17 References [1] Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, no. 1-4, , [2] Crespi G.P.: Proper efficiency and vector variational inequalities, Journal of Information and Optimization Sciences, Vol. 23, No. 1, pp , [3] Crespi G.P., Guerraggio A., Rocca, M.: Minty variational inequality and optimization: scalar and vector case, to appear on the Proceedings of the VII Symposium on Generalized Convexity-Monotonicity, Hanoi, Vietnam, August [4] Crespi G.P., Ginchev I., Rocca M. : Existence of solutions and star-shapedness in Minty variational inequality. J. Global Optim., to appear. [5] Demyanov V.F., Rubinov A.M.: Constructive Nonsmooth Analysis, Peter Lang, Frankfurt am Main, [6] Dentcheva D., Helbig, S.: On variational principles, level sets, well-posedness, and ɛ-solutions in vector optimization. J. Optim. Theory Appl. 89, no. 2, , [7] Dontchev A.L., Zolezzi T.: Well-posed optimization problems,springer, Berlin, [8] Giannessi, F.: Theorems of the alternative, quadratic programs and complementarity problems, in Variational Inequalities and Complementarity Problems. Theory and applications (R.W. Cottle, F. Giannessi, J.L. Lions eds.), Wiley, New York, pp , [9] Giannessi F. : On Minty variational principle, in New Trends in Mathematical Programming, Kluwer, pp , [10] Giannessi F.: On a connection among separation, penalization and regularization for variational inequalities with point to set operators. Rend. Circ. Mat. Palermo (2) Suppl. No. 48, , [11] Ginchev I., Guerraggio A., Rocca M.: From scalar to vector optimization. Submitted [12] Ginchev I, Hoffman A.: Approximation of set-valued functioons by single-valued ones. Discuss. Math. Differ. Incl. Control Optim. 22, no. 1, 33 66, [13] Kinderlehrer, D., Stampacchia, G. (1980): An introduction to variational inequalities and their applications, Academic Press, New York. [14] Hiriart-Urruty J.-B.: New concepts in nondifferentiable programming. Analyse non convexe, Bull. Soc. Math. France 60, 57 85,

18 [15] Hiriart-Urruty J.-B.: Tangent cones, generalized gradients and mathematical programming in Banach spaces. Math. Oper. Res. 4, 79 97, [16] Lee G.M., Kim D.S., Lee B.S., Yen N.D.: Vector variational inequalities as a tool for studying vector optimization problems. Nonlinear Anal. 34, , [17] Minty, G.J.: On the generalization of a direct method of the calculus of variations, Bulletin of American Mathematical Society, Vol. 73, pp , [18] Rubinov, A.M.: Abstract convexity and global optimization, Kluwer, Dordrecht, [19] Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of multiobjective optimization. Mathematics in Science and Engineering, 176. Academic Press, Inc., Orlando, FL, [20] Zaffaroni A.: Degrees of efficiency and degrees of minimality. SIAM J. Optimization, to appear. 9 16